On the procongruence completion of the Teichm\"uller modular group

TL;DR

This paper systematically studies the procongruence completion of the Teichmüller modular group, providing a parametrization of profinite Dehn twists and applications to Galois representations of hyperbolic curves.

Contribution

It introduces a parametrization of profinite Dehn twists in the procongruence completion and describes their centralizers, advancing the understanding of the group's structure.

Findings

Parametrization of profinite Dehn twists.

Description of centralizers of these twists.

Proof of faithfulness of certain Galois representations.

Abstract

For $2g-2+n>0$, the Teichm\"uller modular group $\Gamma_{g,n}$ of a compact Riemann surface of genus $g$ with $n$ points removed $S_{g,n}$ is the group of homotopy classes of diffeomorphisms of $S_{g,n}$ which preserve the orientation of $S_{g,n}$ and a given order of its punctures. Let $\Pi_{g,n}$ be the fundamental group of $S_{g,n}$, with a given base point, and $\hat{\Pi}_{g,n}$ its profinite completion. There is then a natural faithful representation $\Gamma_{g,n}\hookrightarrow Out(\hat{\Pi}_{g,n})$. The procongruence completion $\check{\Gamma}_{g,n}$ of the Teichm\"uller group is defined to be the closure of the Teichm\"uller group $\Gamma_{g,n}$ inside the profinite group $Out(\hat{\Pi}_{g,n})$. In this paper, we begin a systematic study of the procongruence completion $\check{\Gamma}_{g,n}$. The set of profinite Dehn twists of $\check{\Gamma}_{g,n}$ is the closure, inside this group, of the set of Dehn twists of $\GG_{g,n}$. The main technical result of the paper is a parametrization of the set of profinite Dehn twists of $\check{\Gamma}_{g,n}$ and the subsequent description of their centralizers. This is the basis for the Grothendieck-Teichm\"uller Lego with procongruence Teichm\"uller groups as building blocks. As an application, we prove that some Galois representations associated to hyperbolic curves over number fields and their moduli spaces are faithful.